from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,22,27]))
pari: [g,chi] = znchar(Mod(149,2415))
Basic properties
| Modulus: | \(2415\) | |
| Conductor: | \(2415\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2415.dj
\(\chi_{2415}(44,\cdot)\) \(\chi_{2415}(74,\cdot)\) \(\chi_{2415}(149,\cdot)\) \(\chi_{2415}(359,\cdot)\) \(\chi_{2415}(389,\cdot)\) \(\chi_{2415}(494,\cdot)\) \(\chi_{2415}(569,\cdot)\) \(\chi_{2415}(674,\cdot)\) \(\chi_{2415}(704,\cdot)\) \(\chi_{2415}(779,\cdot)\) \(\chi_{2415}(884,\cdot)\) \(\chi_{2415}(914,\cdot)\) \(\chi_{2415}(1019,\cdot)\) \(\chi_{2415}(1124,\cdot)\) \(\chi_{2415}(1229,\cdot)\) \(\chi_{2415}(1514,\cdot)\) \(\chi_{2415}(1859,\cdot)\) \(\chi_{2415}(2039,\cdot)\) \(\chi_{2415}(2144,\cdot)\) \(\chi_{2415}(2384,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((806,967,346,1891)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
| \( \chi_{ 2415 }(149, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(1\) | \(e\left(\frac{47}{66}\right)\) |
sage: chi.jacobi_sum(n)